Please just put something for each question, I don’t care if one or two answers are not correct.
Effect of Variations in Damping Coefficient on the Settling Time of a Mechanical System
Imagine a certain mechanical system with a tunable viscous damper that can be modeled by this
second-order initial value problem:
9y” + αy’ + y = 0; with y(0) = 3 and y'(0) = 6; for α > 0.
1. Settling time at critical damping
a) Find α0, the damping coefficient that results in critical damping.
b) Solve the initial value problem for α = α0, and express the solution in the form y(t) = c1y1 + c2y2.
c) Graph y(t) and find τ(α0), the settling time of the critically-damped system.
2. Settling time of the overdamped system
a) Choose α1 and α2, damping coefficients that result in overdamped systems, with α0 < α1 < α2.
b) Solve the IVP in each case, graph the solutions, and compute settling times τ(α1) and τ(α2)
c) Compare τ(α0), τ(α1), and τ(α2). Do the values make sense? Interpret their relative magnitudes in terms of the mechanical system.
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